In the forthcoming days, I will be summarizing some of the lectures given at the European Science Days summer school in Steyr, Austria. On the first day, there was an interesting lecture by Louis Eeckhoudt about risk and pain disaggregation.
Which lottery would you prefer?
- In lottery A you have a 1/4 chance of getting 0 € and a 3/4 chance of getting 2000 €.
- In lottery B you have a 3/4 chance of getting 1000 € and a 1/4 chance of getting 3000 €.
What did you choose? Be honest…
Most people prefer lottery B. Why? The expected value and variance of the two lotteries are identical. However, the skewness of lottery B is positive, but is negative for lottery A. If people are prudent, they choose lottery B. Mathematically, prudence occurs if the third derivative of the utility function is positive (U”’) is positive. Intuitively, people would rather have an upside risk with small probability than a downside risk with small probability even if the mean and variance of the two lotteries are equal.
Dr. Eeckhoudt also introduced the concept of temperance. An individual is temperate if an an exogenous increase in one risk leads them to reduce risk in other areas. Applying this to alcohol, a temparate person would reduce their wine consumption as their beer consumption increase in order to moderate their aggregate risk of getting drunk. Mathematically, temperate individuals have a utility funciton where the fourth derivative (U””) is negative.
Applications to Health
Dr. Eeckhoudt spoke about the well known phenomenon that risk averse people would like to purchase some sort of insurance. If possible, they will self insure. This differs from self-prevention. Let us look at two examples:
- Self insurance: With probability p, you will become sick and have a utility of x-L(e)-e. With probability 1-p, you will be healthy utility x-e. In this example, by exerting effort, e, you can decrease your health loss L. Thus, the more effort you put forth, the closer will be the two utility levels in each state and thus risk will be dimished.
- Self-prevention: With probability p(e), you will become sick and have a utility of x-L-e. With probability 1-p(e), you will be healthy and have utility x-e. In this example, by exerting effort, e, you can decrease your the probability of becoming sick, p(e). The key insight from Dr. Eeckhoudt was that more prudent people have a lower level of self prevention!
This is certainly an important societal issue since prudence may decrease healthy behaviors such as self-prevention measures against sickness (e.g.: excercising, quitting smoking, immunizations).
Dr. Eeckhoudt underlying message was that if people prefer to disagregate their pain/risk level, rather than combining losses or risk into a single time period or payoff, this has interesting implications. Let us look at prudence in the mutli-dimensional case.
Each individual recieves utility from income, x, and health, h. The variable s represents sickness and e is a zero mean stochastic term which one could interperet as income risk. Which of the following lotteries should people prefer?
- In lottery A you have a 1/2 chance of being healthy and having a risky job (x+e,h) and 1/2 chance of having having a safe job but getting sick (x,h-s).
- In lottery B you have a 1/2 chance being sick and having a risky job (x+e,h-s) and a 1/2 chance of being healthy and having a safe job (x,h).
A prudent person will prefer lottery A to lottery B. In lottery B, sickness and risk are concentrated into one state, where as the risk/losses are disaggregated or spread between the two states.